☰ Contents · Mathematics

Function, monotonicity and extrema

Lessons 25–26 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
26

Monotonicity, greatest and least values of elementary functions

Textbook, Part 2: pp. 8–11
GoalFind the intervals of increase and decrease, the local extrema and the greatest and least values of a function from its graph and from a formula.
New words
increasing function · o‘suvchi funksiyadecreasing function · kamayuvchi funksiyalocal maximum · lokal maksimumgreatest value · eng katta qiymat
Explanation

If for all x₁ < x₂ in an interval I we have f(x₁) < f(x₂), then f is increasing on I; if f(x₁) > f(x₂), it is decreasing. On a graph, moving left to right, the function increases when the ordinates grow and decreases when they fall. On a given interval the greatest value is the largest of all values and the least value is the smallest; they can occur at the ends of the interval or at a turning point of the graph. A local maximum is a point whose value is greater than those near it (a peak); a local minimum is smaller than those near it (a valley). A local maximum need not be the greatest value, since elsewhere the function may be larger still. A linear function y = kx + b increases when k > 0 and decreases when k < 0.

Worked examples
The function y = −2x + 5 is decreasing on [1, 4] (k = −2 < 0). So the greatest value is y(1) = 3 and the least is y(4) = −3.
A graph is given on [−3, 4] with f(−3) = 2, f(0) = 6, f(2) = −1, f(4) = 3 and a smooth curve between these points. It increases on [−3, 0], decreases on [0, 2], increases on [2, 4]. x = 0 is a local maximum (6) and x = 2 is a local minimum (−1). The greatest value is 6 and the least is −1.
Class activity

Record a day’s temperature (for instance outdoor air, measured every two hours with a thermometer, with an adult’s help) in a table, draw the graph and find the intervals of increase and decrease.

Practice
1
When is a function called increasing?
2
Find the greatest value of y = x² on [−3, 2].
3
Find the least value of y = −2x + 5 on [1, 4].
4
Why does y = kx + b increase when k > 0?